[MUSIC]

So our first climate model was too cold and the remedy to that is to add

the greenhouse effect and we're gonna do that in a way that's very, very simple.

You wouldn't understand the greenhouse effect without understanding this

simple model that I'm going to show you.

But of course, reality is much more complicated.

So after we put together this very simple model and understand it,

we'll be spending a fair bit of time in the class actually talking about how to

make it more realistic to make it more like the real world.

So the bare Earth model that we looked at last time was basically this.

It had sunlight coming in and energy leaving according to epsilon,

sigma, temperature of the ground to the 4th power.

So our new model with the greenhouse effect, we're gonna suspend a pane

of glass above the ground and the glass has the property that sunlight can

go through it without any impediment, it's not absorbed by the glass.

But any infrared that comes from the ground is gonna be absorbed by

that pane of glass and the glass itself is going to shine in the infrared

in both directions up and down according to its own temperature.

The temperature of the pane of glass or as I've written the atmosphere,

the temperature of the atmosphere here.

So we're assuming throughout all of this that these epsilon values

are equal to one, so that means that the glass has all of

the oscillators that it needs to make a nice smooth black body curve and

create the full sigma T to the 4th black body spectrum.

Of course, it's not a perfect black body.

Because if it were, it would absorb the sunlight too.

According to the definition, it should absorb and emit all frequencies.

So it's kind of a selective, infrared kind of a black body.

And we're gonna solve for the equilibrium temperature such that

the energy is in steady state, just like we did before.

[SOUND] We're gonna look for the energy in and equate that to the energy going

out and there are multiple places where we can construct a budget in this new model,

because there's actually multiple temperatures.

There's a temperature of the ground and the temperature of the atmosphere.

So starting from the ground, since we are sort of ground based

beings after all, we can just look at the arrows going into and

coming out on the ground and write them as a inputs and outputs.

So what's coming in to the ground here, we've got the same sunlight as before and

we have this arrow of infrared coming down from the sky.

And so that's epsilon, sigma, temperature of atmosphere to the 4th power,

that's what's coming in.

And then what's going out of the ground is just sigma T,

temperature of the ground to the 4th power there.

So that's not really as convenient as the last equation we looked at,

because we have two unknowns in it.

We can't solve for both of them at the same time with just one equation.

If you wanna solve for two unknowns, you need two constraints, two equations.

So here is another one.

This is the budget for the pane of glass.

So we have what comes into the pane of glass is not the solar energy at all.

It is in fact, the epsilon, sigma,

temperature of the ground to the 4th power.

That's this arrow coming into the glass there and

then what's leaving the pane of glass is epsilon, sigma,

temperature of the pane of glass to the fourth power.

And there's this factor of two, because the light is going both upward and

downward.

This two is not the most intuitive thing, you wouldn't probably have thought of it.

But without that factor of two there,

there actually wouldn't be a green house effect as we'll see.

So this is now an equation that has two unknowns in it,

the temperature of the ground and the temperature of the atmosphere,

rather haphazardly written there.

And so we could in principle, take these two equations and

use algebraic substitution to solve for just one of the temperatures.

And then use that, plug that back in to solve for the other one.

And you're welcome to do that, if you like to try your algebra chops, but

there's an easier way to do it.

And it actually has a conceptual benefit,

as well that we can draw a different budget and

it is a budget for the Earth system overall.

So we can draw a line, sort of a boundary

to space above the atmosphere.

And the energy crossing this boundary to space coming down has to balance

the energy crossing the boundary to space going up.

In the steady state, everything has to be in balance.

And so if we write those equations, what's coming down

is just our familiar sunshine L(1-alpha) over 4.

And then what's going out is epsilon, sigma,

temperature of the atmosphere to the 4th.

So this equation should look familiar to you.

This is actually the same equation that we got from the last model for

the temperature of the ground.

Only now, it's that the temperature of the upper layer.

And it turns out, this is sort of a general property of these simple models

and also of the more complicated Earth system that the temperature

where the energy shines out to space is kind of a fulcrum point.

It's fixed by how bright the sunshine is and

how much is reflected away in the albedo.

So this is the naked Earth model and the temperature of the ground is 255 Kelvin.

Here is our greenhouse model with one pane of glass here and

the temperature of the pane of glass is 255 Kelvin again.

In the exercises,

you're gonna work out the balance of this super greenhouse model.

It has two panes of glass, so twice as much greenhouse forcing as that and

what you will find is that the top layer is 255 Kelvin.

And you're also work out a nuclear winter scenario where there's cred in

the atmosphere and so the sunlight gets absorb in the atmosphere,

it doesn't go down to the ground.

And it turns out that it has a big impact on the temperature on the ground but

it has no impact on the top layer, which is again, still 255 Kelvin.

So it's this very useful conceptual thing to keep in mind and

it's also algebraically, much simpler to solve.

Because once you have the top temperature here,

you can take the budget for the atmosphere here and very easily solved for

the temperature of the ground given this temperature of the atmosphere.

It turns out the temperature of the ground is equal of the temperature

of the atmosphere times this factor of the fourth root of 2,

which is about 1.189, so about 20% warmer.

So what's happening is that in our kitchen sink analogy

where the water is at its steady state level in the sink and

enough to drive it down the drain as fast as it's coming in from the faucet.

It's as though, a little piece of carrot came and

got stuck on the drain filter there.

And it doesn't plug it up completely, cuz then the sink would flood and

the analogy would blow up and it would be no good for anybody.

But it sort of partially obstructs it and

it means that the water has to try harder to get out.

And so what happens then is that the water level compensates for that eventually,

not instantaneously, but eventually, it'll build up to a new equilibrium

with a higher water level where the energy flexes are balancing again.

So the water didn't come from a giant bucket full,

the carrot didn't come with a bunch of water in it, nor

did the pane of glass come with a bunch of energy in it.

But it just traps the energy or the water in this case and

allows it to build up to a newer higher concentration.

So if we come back to our table of the Goldilocks planets here,

Venus, Earth and Mars.